Find the value
\[ a = 5 + 2\sqrt{6}, \quad b = \frac{1}{a} \]
\[ b = \frac{1}{5 + 2\sqrt{6}} \times \frac{5 – 2\sqrt{6}}{5 – 2\sqrt{6}} = 5 – 2\sqrt{6} \]
\[ a + b = (5 + 2\sqrt{6}) + (5 – 2\sqrt{6}) = 10 \]
\[ a^2 + b^2 = (a + b)^2 – 2ab \]
\[ ab = 1 \Rightarrow a^2 + b^2 = 10^2 – 2 = 100 – 2 = 98 \]